#6579043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(14,18)(19,28)(20,21)(22,23);(5,7,9,8)(6,10,12,11)(13,14,16,18)(15,22,24,23)(17,20,26,21)(19,27,28,25);(2,4)(5,7,9,8)(6,11,12,10)(13,15,16,24)(14,19,18,28)(17,27,26,25)(20,22,21,23);(5,7,9,8)(6,11,12,10)(13,14,16,18)(15,23,24,22)(17,20,26,21)(19,25,28,27);(5,8,9,7)(6,10,12,11)(14,20)(17,26)(18,21)(19,22)(23,28)(25,27)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250457, #72257035, #72257050, …
maximal normal subgroup of	#72250457, #72257035, #72257050, …
maximal quotient of	#72264598

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486239);